T-048: $s(50)\ge 37/5=7.4$$s(50)\ge 37/5=7.4$

The cases

Results on these cases

15 results in the register on $n=50,51$$n=50,51$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007 · $n=50,51$$n=50,51$

   s(n)≥min(⌈n​⌉,n−2⌊n​⌋+1​+1)$s(n)\ge min(\lceil \sqrt{n}\rceil ,\sqrt{n-2\lfloor \sqrt{n}\rfloor +1}+1)$ for $4\le n\le 324$$4\le n\le 324$

   S3 V0 C1 Lower bound incomplete on these cases, superseded by T-048 and T-090

   Nagamochi · Nagamochi 2005 · source · register

2. 2026-09-04 published T-085 · $n=50,51$$n=50,51$

   Nagamochi 2005, Lemma 1 is false for every container with $a>3$$a>3$ and $b>2$$b>2$

   S3 V3 C3 Correction confirmed

   Karakuş; Vlasenko · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register

3. 2026-09-27 published T-046 · $n=51$$n=51$

   Rectangle-density lower bounds reported for 48 counts in $n=18\dots 95$$n=18\dots 95$

   S3 V0 C0 Lower bound recorded superseded by T-090

   Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · register

4. 2026-09-28 published T-048 this result · $n=50,51$$n=50,51$

   $s(50)\ge 37/5=7.4$$s(50)\ge 37/5=7.4$

   S3 V3 C3 Lower bound confirmed

   Hosono after Daniel, Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point and mixed bounds 2026-09-28 · packet · source · review · register

5. 2026-09-28 published T-070 · $n=51$$n=51$

   Rectangle-density lower bounds replayed at 25 counts in $n=29\dots 95$$n=29\dots 95$

   S3 V3 C3 Lower bound confirmed superseded by T-090

   Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-09-28 · packet · packet · source · review · register

6. 2026-09-29 published T-058 · $n=50,51$$n=50,51$

   Rectangle-certificate ceiling α·UB(n) proved for $n=1$$n=1$..100; B·UB(n) on 64 grid rows

   S2 V3 C3 Method limit confirmed

   Hosono after Tokoharu, Daniel · wand125 tools 2026 · packet · register

7. 2026-09-29 published T-083 · $n=50,51$$n=50,51$

   s(n)≥1/2+n−⌊n​⌋+1/4​$s(n)\ge 1/2+\sqrt{n-\lfloor \sqrt{n}\rfloor +1/4}$ for every nonsquare $8\le n\le 324$$8\le n\le 324$

   S3 V3 C3 Lower bound confirmed on these cases, superseded by T-048 and T-090

   Karakuş · Karakuş 2026 · source · register

8. 2026-10-03 published T-082 · $n=51$$n=51$

   Mixed rectangle-measure lower bounds verified at 22 counts in $n=51\dots 96$$n=51\dots 96$

   S3 V3 C3 Lower bound confirmed on these cases, superseded by T-090

   Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-03 · packet · packet · source · review · register

9. 2026-10-04 published T-090 · $n=51$$n=51$

   Mixed rectangle-measure lower bounds verified at 17 counts in $n=42\dots 96$$n=42\dots 96$

   S3 V3 C3 Lower bound confirmed

   Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-04 · packet · source · review · register

10. 2026-10-05 published T-101 · $n=50,51$$n=50,51$

    Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side

    S2 V3 C3 Upper bound confirmed on these cases, superseded by T-126 and T-155

    Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register

11. 2026-10-08 published T-125 · $n=51$$n=51$

    Complete rational construction reports at 25 counts

    S3 V3 C3 Upper bound confirmed superseded by T-126

    Xu · ry-xu square packing 2026 · packet · register

12. 2026-10-08 published T-126 · $n=51$$n=51$

    Undilated n51 construction over Q(sqrt2)

    S3 V3 C3 Upper bound confirmed

    Xu · ry-xu square packing 2026 · packet · register

13. 2026-10-12 established T-155 · $n=50$$n=50$

    Exact regularized sides at 59 counts, 5.4e-20 to 9.9e-15 below the ceilings they replace

    S1 V3 C3 Upper bound confirmed

    Levy after Daniel · register

14. 2026-10-13 published T-180 · $n=50,51$$n=50,51$

    Closed-form lower bound $s(N)\ge a(N)$$s(N)\ge a(N)$ for every non-square $N\ge 49$$N\ge 49$, reported

    S4 V0 C1 Lower bound reviewed on these cases, superseded by T-048 and T-090

    Hosono · wand125 closed form 2026-10-14 · packet · packet · register

15. 2026-10-16 published T-204 · $n=51$$n=51$

    Mixed rectangle-measure lower bound $s(51)\ge 3011/400$$s(51)\ge 3011/400$ reported, on a 2,073-direction declared net

    S3 V0 C0 Lower bound recorded

    Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register